Question

Suppose a sphere, cylinder, and cone all share the same radius. The cylinder and cone also share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone?(1 point) Responses cylinder = sphere – cone cylinder = sphere – cone sphere = cylinder + cone sphere = cylinder + cone cone = cylinder – sphere cone = cylinder – sphere cone = sphere – cylinder

Answer 1

To compare the volume formulas of a sphere, cylinder, and cone with the given dimensions, let's first recall the volume formulas for each shape:

1. **Sphere**: \( V_{\text{sphere}} = \frac{4}{3} \pi r^3 \)
2. **Cylinder**: \( V_{\text{cylinder}} = \pi r^2 h \)
3. **Cone**: \( V_{\text{cone}} = \frac{1}{3} \pi r^2 h \)

Given that the radius \( r \) is the same for all shapes, and the height \( h \) of the cylinder and cone is twice the radius \((h = 2r)\), we can substitute \( h \) with \( 2r \) in the volume formulas for the cylinder and cone:

- **Cylinder**:
\[
V_{\text{cylinder}} = \pi r^2 (2r) = 2 \pi r^3
\]

- **Cone**:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now, let’s establish the relationships:

1. **Cylinder and Cone**:
\[
V_{\text{cylinder}} = 2 \pi r^3
\]
\[
V_{\text{cone}} = \frac{2}{3} \pi r^3
\]

2. **Sphere**:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]

Considering these formulas, we can determine if there is a relationship among them. Let's examine a few combinations:

- **Addition of Cylinder and Cone’s volume**:
\[
V_{\text{cylinder}} + V_{\text{cone}} = 2 \pi r^3 + \frac{2}{3} \pi r^3 = 2 \pi r^3 + \frac{2}{3} \pi r^3 = \left(2 + \frac{2}{3}\right) \pi r^3 = \frac{6}{3} \pi r^3 + \frac{2}{3} \pi r^3 = \frac{8}{3} \pi r^3
\]

- This isn’t equal to \(V_{\text{sphere}} \):
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]

- **Comparison of Cone and Cylinder’s volume**:
\[
V_{\text{cone}} = \frac{2}{3} \pi r^3
\]
\[
V_{\text{cylinder}} - V_{\text{sphere}} = 2 \pi r^3 - \frac{4}{3} \pi r^3 = \left(2 - \frac{4}{3}\right) \pi r^3 = \frac{6}{3} \pi r^3 - \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]

From this, it’s clear:
\[
V_{\text{cone}} = V_{\text{cylinder}} - V_{\text{sphere}}
\]

Thus, the correct relationship is:
\[
\boxed{ \text{cone} = \text{cylinder} - \text{sphere}}
\]